Mathematics 10 — Exercise 2.1 · Q4 (ii)

Full question

Use the quadratic formula to solve the following equation: 2 × x²-7 × x-15 = 0.

Step-by-step solution

1. Substitute the function expressions, keeping the brackets.

   $$--2\times x^2+7\times x+15-0$$

2. Remove the brackets. A preceding minus sign reverses every sign inside that bracket.

   $$2\times x^2-7\times x-15$$

3. Write EVERY distributed product before collecting terms.

   $$-15-7\times x+2\times x^2-0$$

4. Calculate this individual product, including its sign and powers.

   $$-15=-15$$

5. Calculate this individual product, including its sign and powers.

   $$-7\times x=-7\times x$$

6. Calculate this individual product, including its sign and powers.

   $$2\times x^2=2\times x^2$$

7. Calculate this individual product, including its sign and powers.

   $$\left(-1\right)\times 0=0$$

8. These terms have the same variable part. Add ONLY their coefficients.

   $$-15+0=-15$$

9. These terms have the same variable part. Add ONLY their coefficients.

   $$-7=-7$$

10. These terms have the same variable part. Add ONLY their coefficients.

    $$2=2$$

11. Put the simplified groups together.

    $$2\times x^2-7\times x-15$$

12. Collect terms with the same power and simplify their coefficients.

    $$2\times x^2-7\times x-15$$

13. Expand the brackets and subtract the right-hand terms from both sides. Collect powers of the variable.

    $$2\times x^2-7\times x-15=0$$

14. Compute the discriminant one operation at a time: square b, multiply 4,a,c, then subtract.

    $${\left(-7\right)}^2-8\times \left(-15\right)$$

15. Calculate this operation before combining the remaining terms.

    $${\left(-7\right)}^2=49$$

16. Calculate this operation before combining the remaining terms.

    $$\left(-4\right)\times 2\times \left(-15\right)=120$$

17. Calculate this operation before combining the remaining terms.

    $$49+120=169$$

18. Compare with ax² + bx + c = 0 and read a, b, c WITH their signs.

    $$(a=2,b=-7,c=-15)$$

19. Compute D = b² − 4ac. A minus times a minus becomes a plus.

    $$D={\left(-7\right)}^2-8\times \left(-15\right)$$

20. Simplify the discriminant.

    $$D=169$$

21. Use x = (−b ± √D)/(2a). Show the plus and minus cases separately.

    $$(x=\frac{7+13}{4},x=\frac{7-13}{4})$$

22. Simplify each numerator and divide by 2a.

    $$\{-\frac{3}{2},5\}$$

23. Calculate this root branch separately, preserving the sign.

    $$x=\frac{7+1\times \sqrt{169}}{2\times 2}$$

24. Calculate this operation before combining the remaining terms.

    $$\sqrt{169}=13$$

25. Calculate this operation before combining the remaining terms.

    $$1\times 13=13$$

26. Calculate this operation before combining the remaining terms.

    $$7+13=20$$

27. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

28. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{20\times 1}{1\times 4}$$

29. Calculate these two products separately.

    $$\frac{20}{4}$$

30. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=4$$

31. Show the two divisions explicitly.

    $$\frac{\frac{20}{4}}{\frac{4}{4}}$$

32. Evaluate the numerator division.

    $$\frac{20}{4}=5$$

33. Evaluate the denominator division.

    $$\frac{4}{4}=1$$

34. Write the reduced fraction.

    $$5\times \frac{1}{1}$$

35. Calculate this operation before combining the remaining terms.

    $$20\times \frac{1}{4}=5$$

36. Calculate this root branch separately, preserving the sign.

    $$x=\frac{7-\sqrt{169}}{2\times 2}$$

37. Calculate this operation before combining the remaining terms.

    $$\sqrt{169}=13$$

38. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 13=-13$$

39. Calculate this operation before combining the remaining terms.

    $$7-13=-6$$

40. Calculate this operation before combining the remaining terms.

    $$2\times 2=4$$

41. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-6\right)\times 1}{1\times 4}$$

42. Calculate these two products separately.

    $$-\frac{6}{4}$$

43. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=2$$

44. Show the two divisions explicitly.

    $$\frac{-\frac{6}{2}}{\frac{4}{2}}$$

45. Evaluate the numerator division.

    $$-\frac{6}{2}=-3$$

46. Evaluate the denominator division.

    $$\frac{4}{2}=2$$

47. Write the reduced fraction.

    $$-\frac{3}{2}$$

48. Calculate this operation before combining the remaining terms.

    $$\left(-6\right)\times \frac{1}{4}=-\frac{3}{2}$$

49. Substitute this root into the original equation before simplifying.

    $$0=0$$

50. Calculate both sides; their equal values confirm the root.

    $$0=0$$

51. Substitute this root into the original equation before simplifying.

    $$0=0$$

52. Calculate both sides; their equal values confirm the root.

    $$0=0$$

Answer

The final answer is shown below.